Supporting Information for “Resilience, invariability,
and ecological stability across levels of organization”
Bart Haegeman, Jean-François Arnoldi, Shaopeng Wang, Claire de Mazancourt,
José M. Montoya & Michel Loreau
Supporting texts
Text S1 Definition of stability measures
Text S2 Details of consumer-resource model (Figs. 2 and 3)
Text S3 Effect of absent species on stability measures
Text S4 Details of random interactions model (Fig. 4)
Supporting figures
Figure S1 Invariability close to extinction threshold
Figure S2 Details of resource competition model
Figure S2 Effect of absent species in resource competition model
Figure S3 Effect of absent species in random interactions model
1
S1 Definition of stability measures
In this section we define the stability measures used in the main text. We consider a
continuous-time dynamical system,
dXi
= Fi (X1 , . . . , Xn )
dt
for i = 1, . . . , n,
or in vector form,
dX
= F (X),
(1)
dt
where X is a vector containing the dynamical variables Xi and F is a non-linear map
consisting of the right-hand sides Fi of the dynamical equations.
We assume that the stationary state of dynamical system (1) is an equilibrium point (that
is, we exclude limit cycles and chaotic attractors). Denoting the equilibrium point by X ∗ ,
we have F (X ∗ ) = 0. Linearizing around this equilibrium point, we get the linear system
dX̃
= AX̃,
dt
(2)
with X̃ = X − X ∗ and A the Jacobian matrix of F evaluated at X ∗ ,
∂Fi (X) .
Aij =
∂Xj X=X ∗
The condition for stability of the linear dynamical system (2), and thus the condition for
local stability of the equilibrium point X ∗ , reads that all eigenvalues of A have negative
real part,
<e(λi ) < 0
for all eigenvalues λi of A.
Equivalently, the stability condition can be expressed in terms of the dominant eigenvalue
λdom of dynamical system (2), that is, the eigenvalue of A with the largest real part,
<e(λdom ) < 0.
(3)
From here on we assume X ∗ to be stable. We are interested in quantifying how stable the
equilibrium point X ∗ is. We define several such stability measures.
Asymptotic resilience R
Looking at condition (3), a straightforward choice for a stability measure is
R = −<e(λdom ).
(4)
Clearly, R is positive for stable systems. It is close to zero when the dynamical system is
close to the stability boundary (3).
Measure R is often called asymptotic resilience, because, generically, it equals the asymptotic rate of return to equilibrium after a pulse perturbation. This property can be easily
formulated in the case of a single, real dominant eigenvalue (other cases can be dealt with
similarly, but with additional technical difficulties). Denote by U the initial displacement
P
of the system, and assume that a linear combination i Wi X̃i (t) of the dynamical variables
is observed. Then, the observed return to equilibrium is
D(t) = W T X̃(t) = W T eAt U.
2
We impose a genericity condition: displacement vector U should not be orthogonal to the
left eigenvector associated with λdom , and observation vector W should not be orthogonal
to the (right) eigenvector associated with λdom . Then,
R = lim −
t→∞
1 d|D(t)|
ln |D(t)|
= lim −
,
t→∞
|D(t)| dt
t
that is, the long-term rate of return to equilibrium is given by asymptotic resilience. Note
that the genericity condition is almost always satisfied, so that the long-term return rate is
essentially independent of the applied perturbation and of the observed system variable. In
the main text this property is referred to as the rigidity of asymptotic resilience.
Invariability measures
Alternatively, stability can be quantified by looking at the response of dynamical system (1)
to perturbations. We introduce a stochastic forcing term that we assume to be sufficiently
small, so that we can study its effects using the linear dynamical system (2). The resulting
dynamical system is
X
dX̃i
=
Aij X̃j + γi (t)
dt
j
for i = 1, . . . , n,
(5)
with γi (t) the stochastic forcing. We make the following assumptions:
• γi (t) is a Gaussian variable with zero mean. This guarantees that the deterministic
behavior of dynamical system (5) is given by dynamical system (2). Furthermore, we
assume that the variance of γi (t) is proportional to (Xi∗ )2 . This corresponds to the
assumption that in the nonlinear dynamical system the stochastic forcing acts in a
multiplicative way. Alternatively, one could assume, for example, that the variance of
γi (t) is proportional to Xi∗ , which would lead to quite different invariability measures.
• γi (t) as a function of t (and for fixed i) is white noise. This means that the γi (t)
at different t are mutually independent and have the same Gaussian distribution.
Equivalently, this means that the frequency spectrum of the noise is colorless, that is,
all frequency components have the same (expected) amplitude.
• γi (t) for different i (and for fixed t) are independent. This means that there are no
correlations between the noise terms acting on different system variables. Without
detailed information about the perturbations, correlations between γi (t) and γj (t)
can be both positive and negative. Hence, the assumption of no correlations can be
considered as a mean-case scenario (worst-case scenarios are considered by Arnoldi
et al. 2016).
Based on these assumptions, we can write dynamical system (5) using the formalism of
stochastic differential equations (Van Kampen, 2007),
X
dX̃i =
Aij X̃j dt + σ Xi∗ dWi (t)
for i = 1, . . . , n,
j
or in vector form,
dX̃ = AX̃dt + σ diag(X ∗ ) dW (t),
3
(6)
with W (t) a vector of n independent Wiener processes Wi (t). Dynamical system (6) can be
simulated using the iterative scheme
√
X̃(t + ∆t) = X̃(t) + AX̃(t)∆t + σ diag(X ∗ ) ξ(t) ∆t,
for a small time step ∆t. Here the vector ξ(t) contains n independent Gaussian variables
with zero mean and unit standard deviation. The constant σ is a measure of the intensity
of the stochastic forcing. For small σ the linearization provides an accurate approximation
of the nonlinear dynamics (Fig. S1). Note that σ 2 can be interpreted as the variance per
unit of time of the applied stochasticity, implying that σ 2 has units of 1/time.
For stable systems, dynamics (6) approach a stationary state after a sufficiently long time.
In the stationary state dynamical variables X̃i (t) fluctuate around zero (and dynamical
variables Xi (t) fluctuate around Xi∗ ). The covariance matrix C of the stationary fluctuations
can be obtained from the Lyapunov equation (Van Kampen, 2007),
A C + CAT + σ 2 diag(X ∗ )2 = 0.
(7)
The covariance matrix C allows us to compute the stationary fluctuations of any system
variable Q(X). Recall that the linearization is based on the assumption that the deviations X̃ from the equilibrium X ∗ are small (which is satisfied for small σ). We use the same
assumption to restrict attention to variables Q(X) that are of first order in X̃,
Q(X) = Q(X ∗ ) + VQT X̃,
where VQ is the gradient vector of Q evaluated at X ∗ ,
∂Q(X) VQ i =
.
∂Xi X=X ∗
Then, the variance of the stationary fluctuations are given by
Var(Q) = VQT C VQ .
By normalizing this variance with respect to the mean Q(X ∗ ) of variable Q(X), and with
respect to the intensity σ of the applied perturbation, we obtain the variability measure
Var(Q)/Q(X ∗ )2 /σ 2 . The invariability IQ of system variable Q is then defined as
IQ =
σ2
1
σ 2 Q(X ∗ )2
=
,
2 CV(Q)2
2 Var(Q)
(8)
where CV denotes the coefficient of variation. The factor 1/2 can be justified by looking at
one-dimensional dynamical systems. Setting n = 1, the linearized system (2) becomes
dX̃
= −aX̃,
dt
where a is a positive number (assuming that the system is stable). Clearly, R = a. We
compute the stationary fluctuations of a linear system variable Q(X) = qX. Solving the
Lyapunov equation (7), we find that
C=
σ2 X ∗ 2
2a
and
Var(Q) =
q2 σ2 X ∗ 2
,
2a
so that,
IQ =
σ 2 Q(X ∗ )2
σ2 q2 X ∗ 2
= a = R.
=
2 Var(Q)
2 q2 σ2 X ∗ 2
2a
4
Hence, the factor 1/2 in definition (8) guarantees that invariability IQ and resilience R coincide for one-dimensional dynamical systems. This equality indicates that, more generally,
the values of invariability and resilience can be compared, and in particular, that they have
the same units, namely, units of 1/time. Note that, while invariability is closely related to
the coefficient of variation (which is unitless), the comparability with asymptotic resilience
is due to its normalization with respect to the applied stochasticity σ 2 (which has units of
1/time).
Ecosystem invariability Ieco
The examples of the paper deal with dynamical systems of which dynamical variables Xi
represent biomass. To quantify the stability at the ecosystem level, we consider the invariability of the total biomass present in the ecosystem. As system variable Q we take total
P
biomass, that is, Q(X) = Xtot = i Xi . Then, the corresponding invariability Ieco is
Ieco =
∗ 2
σ 2 (Xtot
σ2
1
)
=
2
2 CV(Xtot )
2 Var(Xtot )
(9)
We call this measure ecosystem invariability. By construction, it captures (an aspect of)
stability at the ecosystem level. Note that, in contrast to R, Ieco need not be close to zero
when the dynamical system is close to the stability boundary (3).
Population invariability Ipop
Invariability can also be used to address stability at the population level. Taking as system
variable Q the biomass of a single population, Q(X) = Xi , we get
IXi =
σ 2 (Xi∗ )2
2 Var(Xi )
where Xi is a biomass variable.
We combine these invariabilities into a single, ecosystem-wide measure of population-level
stability. There are different possibilities to do so. One could start from the idea that the
most unstable population determines the population-level stability of the entire ecosystem.
This leads to the definition of minimum invariability,
Imin = min IXi =
i
σ2
(Xi∗ )2
min
.
2 i Var(Xi )
(10)
By construction, measure (10) potentially depends on the invariability of rare species. In
fact, as we show in the paper, minimum invariability is typically determined by rare species.
This might be problematic, especially from an estimation point of view. To estimate a
measure that depends strongly on rare species, we need accurate data for these rare species,
which is typically difficult to obtain, precisely because they are rare.
Alternatively, one could consider the average invariability of all species.
(Lehman & Tilman, 2000),
1X
σ 2 X (Xi∗ )2
IXi =
,
n i
2n i Var(Xi )
For example
(11)
where n denotes the number of species (that is, the number of terms in the sum). However,
also for measure (11) rare species can have a relatively large effect. There are two reasons
5
for this. First, IXi need not be small when Xi∗ is small. It is even possible that the
contribution of a rare species dominates the sum in (11). Second, even if all rare species
have small invariability, their presence will reduce the average invariability (11). If there
are many rare species, this can have a large effect on the outcome of (11).
These problems can be solved by using a weighted average (weighted by population biomass).
Here we propose to take a weighted average of the coefficients of variation (Thibaut & Connolly, 2013),
∗ 2
Xtot
σ2
1
σ2
Ipop =
(12)
2 =
2
2 P Xi∗
2 P p
CV(X
)
Var(X
)
∗
i
i
i X
i
tot
which we call population invariability. The last expression in (12) shows that rare species can
only contribute to this stability measure if their summed standard deviation is comparable to
those of common species, which is improbable. Moreover, Ipop has the additional advantage
that it directly connects to Ieco . Indeed, their ratio equals
Var(Xtot )
Ipop
= P p
2 ,
Ieco
Var(X
)
i
i
a quantity that has been proposed as a measure of ecosystem-wide synchrony (Loreau &
de Mazancourt, 2008). It is always smaller than or equal to one, and reaches the maximal
value of one for a perfectly synchronous ecosystem. Note that, just as Ieco but contrary
to R, Ipop need not be close to zero when the dynamical system is close to the stability
boundary (3).
S2 Details of consumer-resource model (Figs. 2 and 3)
In Fig. 2 we analyze the stability properties of a consumer-resource model. This model has
a pool of S = 3 consumer species competing for R = 2 resources. We use the variable Ni
for the biomass of consumer species i and the variable Rk for the amount of resource k.
Resource Rk is consumed by consumer Ni at rate cik and transformed into biomass with
efficiency eik . All consumers Ni have mortality rate m. We consider abiotic resources with
supply rate ak and supply level Ak . This leads to the dynamical equations,
X
dNi
=
cik Rk Ni − mNi ,
i = 1, 2, 3,
dt
k
X
dRk
= a(Ak − Rk ) −
cik Rk Ni ,
k = 1, 2.
dt
i
(13)
The coexistence diagram corresponding to Fig. 2 is shown in Fig. S1. The zero net growth
isoclines of the consumer species intersect, defining coexistence regions in the plane of
resource supply levels (A1 , A2 ). In Fig. 2 the resource supply levels Ak are varied such that
regions of competitive exclusion alternate with regions of coexistence (Fig. S1, black thin
solid line). The other parameter values of model (13) are m = 1, a = 0.6 and
c11 = 0.2790
c21 = 0.7071
c31 = 0.9409
c12 = 0.9603
c22 = 0.7071
c32 = 0.3387.
In Fig. 3 we study the stability properties of a reduced model obtained from model (13).
To constuct the reduced model, we combine the individual consumer species into a single
6
aggregate consumer variable Nagg . Denoting the aggregate functional response by fk (Rk ),
the dynamical equations are
X
dNagg
=
fk (Rk )Nagg − mNagg ,
dt
k
(14)
dRk
= a(Ak − Rk ) − fk (Rk )Nagg ,
k = 1, 2.
dt
Whereas the original model (13) describes the population-level structure of the consumer
guild, the reduced model (14) can be seen as an ecosystem-level model in which the total
biomass of the consumer guild is lumped.
To construct the functional response fk (Rk ) of the reduced model (14), note that the species’
consumption rates in the original model satisfy a trade-off: a species with a large consumption rate of resource 1 has a small consumption rate of resource 2 (and vice versa). In fact,
the consumption rates were chosen to lie on the trade-off curve c2i1 + c2i2 = m. Then, we
consider the three species of the original model as members of a continuous pool of consumer species, spanning the entire trade-off curve. We assume that in the reduced model
the aggregate consumer has the consumption characteristics of the best competitor (for
given resource supply levels A1 and A2 ) from this continuous pool of consumer species.
This assumption allows us to derive the reduced functional response fk (Rk ).
Explicitly, we replace the three consumers Ni by a pool of consumers N (x), indexed by
√
the continuous parameter x ∈ [ 0, m ]. Consumer x has consumption rates c1 (x) = x and
√
c2 (x) = m − x2 for resource 1 and 2, respectively. The corresponding zero net growth
isocline is given by c1 (x)R1 + c2 (x)R2 = m. There is a single resource point (R1 , R2 ) for
which consumer x is the best consumer; it is given by R1 = c1 (x) and R2 = c2 (x). Indeed,
this point lies on the zero net growth isocline of consumer x and below (i.e., closer to the
origin) the zero net growth isoclines of any other consumer. There is a half-line of resource
supply points (A1 , A2 ) leading to this resource point at equilibrium. In this equilibrium
consumer x excludes the other consumers, so that Nagg = Ni . Its per capita consumption
rates are c1 (x)R1 = c1 (x)2 = R12 for resource 1 and c2 (x)R2 = c2 (x)2 = R22 for resource 2.
Hence, we find that fk (Rk ) = Rk2 .
We note that this model reduction is not intended as a generally applicable technique.
Simply, in this specific example, it allows us to link ecosystem-level model (14) to populationlevel model (13).
The zero net growth isocline of the aggregate consumer is plotted as a black thick dashed
line in Fig. S1. It lies below the zero net growth isoclines of the three consumer species, and
is tangent to each of them. This graphically illustrates the construction of the aggregate
functional response. The other parameter values of model (14) are m = 1 and a = 1. The
resource supply levels Ak are varied along the black thin full line in Fig. S1.
S3 Effect of absent species on stability measures
We define an absent species as a species that belongs to the species pool, but that does not
persist in the equilibrium community. Here we describe how the stability patterns of Fig. 2
dependent on the way absent species are dealt with. We show that asymptotic stability R
is strongly depend on absent species, while invariabilities Ipop and Ieco are independent of
absent species.
7
Suppose species i is absent in the equilibrium community, Xi∗ = 0. Then, dynamical
equation (1) of species i can be written as
dXi
= Fi (X) = Xi Gi (X).
dt
Using this notation, we find that row i of Jacobian matrix A is equal to

G (X ∗ ) if j = i
i
Aij =
0
if j 6= i.
All components of row i are zero except the diagonal component. As a consequence, the
eigenvalues of A are equal to this diagonal component together with the eigenvalues of a
smaller matrix A(i) . This matrix A(i) is obtained by eliminating row i and column i of
matrix A. If matrix A has dimensions n × n, matrix A(i) has dimensions (n − 1) × (n − 1).
As a result, the stability condition (3) decomposes into two parts: matrix A(i) should be
stable (that is, all its eigenvalues should have negative real part) and diagonal component
Aii should be negative. The ecological interpretation is as follows. First, note that the
matrix A(i) is obtained by eliminating species i from the species pool. Hence, the first part
of the stability condition states that the other species should constitute a stable equilibrium, thereby discarding their interactions with species i. Second, note that the diagonal
component Aii = Gi (X ∗ ) is equal to the invasion fitness of species i in the equilibrium
community of the other species. To see this, suppose that variables Xj for all j 6= i are at
equilibrium, Xj = Xj∗ , and that variable Xi is very small but different from zero. Then, the
∗
i
relative rate of change X1i dX
dt of variable Xi is equal to Gi (X ). Clearly, species i, which is
absent in the equilibrium community, should be unable to invade for the equilibrium to be
stable.
This implies that absent species can have a large effect on asymptotic resilience R. Indeed, if an absent species is almost able to invade the equilibrium community (that is, a
negative invasion fitness that is small in absolute value), its invasion fitness can determine
the dominant eigenvalue λdom . Hence, the choice of whether to include absent species in
the computation of asymptotic resilience can change the resulting stability pattern qualitatively. In particular, when approaching a bifurcation point in which an absent species
becomes able to invade, asymptotic resilience drops to zero when accounting for that absent
species, but stays positive otherwise (Fig. S2; compare panels a and d).
Similarly, we can analyze the effect of absent species on invariabilities Ipop and Ieco . To
do so, we solve the Lyapunov equation (7) for an equilibrium in which species i is absent,
Xi∗ = 0. Then, as explained above, row i of Jacobian matrix A has zero components except
on the diagonal. We compute component (i, i) of the Lyapunov equation,
X
X
2
0=
Aik Cki +
Cik Aik + σXi∗ = 2Aii Cii .
k
k
Hence, because Aii 6= 0, we find that Cii = 0. Recalling that C is a covariance matrix, we
2
know that Cjk
≤ Cjj Ckk for all j and k (Cauchy-Schwarz inequality). This implies that
Cik = Cki = 0 for all k. Hence, the non-zero components of the covariance matrix C can be
obtained by solving the Lyapunov equation for the dynamical system in which the absent
species have been removed from the species pool. This implies that invariabilities Ipop and
Ieco are insensitive to the inclusion of absent species.
8
The different effects of absent species on the stability measures have important consequences
for linking these stability measures (Fig. S2). When not accounting for absent species,
that is, when restricting Jacobian matrix A to persistent species, there is a qualitative
link between asymptotic resilience R and minimum invariability Imin (Fig. S2; compare
panels a and c). When accounting for absent species, that is, when computing Jacobian
matrix A for the entire species pool, asymptotic resilience R changes while single-population
invariabilities IXi do not. Hence, the qualitative link between them is lost. However,
this link can be recovered by introducing the invasion fitness Gi (X ∗ ) of absent species i.
Explicitly, we take the minimum of the set
S
IXi species i is present
− Gi (X ∗ ) species i is absent .
By doing so, we get again a qualitative link between asymptotic resilience and singlepopulation invariabilities (Fig. S2; compare panels d and f).
It may seem artificial, when computing the stability of an ecosystem, to include species
that are absent from the ecosystem. However, one should note that the previous models
are based on the (unrealistic) assumption that the modelled ecosystem is closed. When
introducing a weak immigration flow from the species pool, all species are present in the
ecosystem. In particular, those that were absent in the closed system, are present at a small
abundance in the open system. Then, the stability patterns for asymptotic resilience are
dissimilar to those when accounting only for persistent species, but they are similar to those
when accounting also for absent species (Fig. S2; compare panel g to panels a and d). This
suggests that the patterns obtained for the entire species pool are the relevant ones, as only
those are robust with respect to the addition of a small amount of immigration. Note that
invariabilities Ipop and Ieco (which are insensitive to the inclusion of absent species) are
also robust with respect to a weak immigration flow.
Finally, we remark that absent species can also affect the generic link between asymptotic
resilience and the long-term rate of return to equilibrium. When asymptotic resilience is
determined by an absent species i, the return rate for any other species k 6= i is different
from (and larger than) R. Mathematically, this is due to the particular structure of the left
eigenvector ϕdom associated with the dominant eigenvalue λdom . Indeed, all components
k 6= i of ϕdom are zero, as can be seen from the structure of the Jacobian A described
above. More generally, when asymptotic resilience is determined by species i with non-zero
but small equilibrium abundance, the associated left eigenvector ϕdom typically has small
components k 6= i compared to component i. This phenomenon, called localization, reduces
the relevance of asymptotic resilience as a dynamical characteristic (Suweis et al., 2015).
S4 Details of random interactions model (Fig. 4)
Here we describe the details of the model with random competitive interactions used in
Fig. 4. It is based on the Lotka-Volterra dynamical equations,
Spool
X
dNi
= Ni ai −
bij Nj ,
dt
j=1
i = 1, . . . , Spool ,
with ai = 1 and bii = 1. The interaction coefficients bij , i 6= j are obtained by randomly
drawing numbers from the Gaussian distribution with mean 0 and standard deviation 0.2,
9
and taking the absolute value. In the resulting dynamical system, the equilibrium in which
all species coexist is not necessarily feasible and stable Therefore, we integrate the dynamical
equations until reaching a new equilibrium, which is feasible and stable (note that all our
simulations eventually converged to an equilibrium). For this equilibrium, which has S ≤
Spool species, we compute the three stability measures R, Ipop and Ieco . We repeat this
procedure for values of Spool in the range from 1 to 100 until our data set consists of 1000
equilibria for each value of S in the range from 1 to 30.
In Fig. S3 we illustrate the robustness of the patterns shown in Fig. 4 with respect to absent
species. Absent species enter the patterns in two different ways. First, we can compute
Jacobian matrices A with or without absent species. As explained in Appendix S3, this
choice has no effect on invariabilities Ipop and Ieco , but it does affect asymptotic resilience
R. Second, we can plot the stability measures against the number of species present in the
equilibrium community (that is, excluding absent species), or against the number of species
in the species pool (that is, including absent species). This choice affects all stability
measures. Fig. S3 shows that the stability patterns are robust with respect to both choices.
References
Arnoldi, J.F., Loreau, M. & Haegeman, B. (2016). Resilience, reactivity and variability: a
mathematical comparison of ecological stability measures. J. Theor. Biol., 389, 47–59.
Lehman, C.L. & Tilman, D. (2000). Biodiversity, stability, and productivity in competitive
communities. Am. Nat., 156, 534–552.
Loreau, M. & de Mazancourt, C. (2008). Species synchrony and its drivers: neutral and
nonneutral community dynamics in fluctuating environments. Am. Nat., 172, E48–E66.
Suweis, S., Grilli, J., Banavar, J.R., Allesina, S. & Maritan, A. (2015). Effect of localization
on the stability of mutualistic ecological networks. Nature Comm., 6, 10179.
Thibaut, L.M. & Connolly, S.R. (2013). Understanding diversity-stability relationships:
towards a unified model of portfolio effects. Ecol. Lett., 16, 140–150.
Van Kampen, N.G. (2007). Stochastic Processes in Physics and Chemistry. Elsevier.
10
Figure S1
System response (stdev)
X ∗ = a = 1.0
0
X ∗ = a = 0.1
(a)
10
X ∗ = a = 0.01
−1
(b)
(c)
10
−1
10
−1
−2
10
10
−2
10
−2
−3
10
10
−3
10
−3
−4
10
10
−3
10
−2
10
−1
10
Applied perturbation σ
0
10
−3
10
−2
10
−1
10
Applied perturbation σ
0
10
−3
10
−2
10
−1
10
0
10
Applied perturbation σ
Figure S1: Invariability close to extinction threshold. Logistic population model dX/dt =
aX − bX 2 with b = 1.0 fixed and (a) a = 1.0; (b) a = 0.1; (c) a = 0.01. The population
becomes extinct (equilibrium biomass X ∗ tends to zero) when the bifurcation parameter a
tends to zero. We apply a white-noise perturbation with intensity σ (on x-axis), and measure
p
the fluctuations of the system response by the standard deviation Var(X) (on y-axis). Red
line with crosses: nonlinear model; green line with circles: linearization around equilibrium.
Results for nonlinear and linear model coincide for weak perturbations, but they start to
differ at smaller intensity σ when approaching the bifurcation. Nevertheless, however close
to the bifurcation, one can always take σ small enough such that the linearization provides
an accurate approximation of the nonlinear dynamics.
11
Figure S2
3
Resource level 2
Sp1
Sp2
2
1
Sp3
0
0
1
2
3
Resource level 1
Figure S2: Details of resource competition model. Colored lines: zero net growth isoclines
of the three consumer species (a species can persist at equilibrium if resource levels (R1 , R2 )
lie on its zero net growth isocline). Grey-shaded triangles: coexistence regions (two species
can coexist at equilibrium if resource supply levels (A1 , A2 ) lie in their coexistence region).
Black thin solid line: range of resource supply levels (A1 , A2 ) scanned along the x-axis of
Figs. 2 and 3. Black thick dashed line: zero net growth isocline of aggregate consumer in
reduced model.
12
Figure S3
(b) Sp1
0.4
0
1
2
0
0
1
2
2
Invariability biomass
0.1
0
1
2
Resource supply
3
0
1
2
1
2
3
1
2
3
1
2
3
0.2
0.1
0
3
0
(i)
0.4
0.2
0
0
0
0
(h)
0.2
0.2
(f)
0.2
0
0.4
3
0.4
3
(g)
Asymptotic resilience
1
Minimum invariability
Invariability biomass
Asymptotic resilience
0.1
0
0
(e)
0.2
0
0.2
3
(d)
(c)
Sp3
Minimum invariability
0
Sp2
0.4
Minimum invariability
Invariability biomass
Asymptotic resilience
(a)
0.8
1
2
Resource supply
3
0.2
0.1
0
0
Resource supply
Figure S3: Effect of absent species in resource competition model. Same model as Fig. 2.
(a) When we restrict the system to species present in the equilibrium community, asymptotic resilience is discontinuous each time a species is able to invade (or a species becomes
extinct). (b & c) The same pattern is obtained by taking the minimum of single-population
invariabilities (shown for the three species in panel b; their minimum is shown in panel c).
(d) When we account for absent species, asymptotic resilience is continuous, but drops to
zero at each bifurcation corresponding to an invasion or an extinction. (e & f) This pattern
can be reproduced by taking the minimum of single-population invariability (for persistent
species; full line in panel e) and (the opposite of) invasion fitness (for absent species; dashed
line in panel e). (g) When adding a small immigration flow, asymptotic resilience has a
behavior similar to the one obtained for the entire species pool. Strictly speaking, there
are no longer invasion/extinction bifurcations, but these bifurcations (for the case without
immigration) are still visibile in the stability pattern. (h & i) The same pattern is obtained
by taking the minimum of single-population invariabilities. Note that all species are present
in the equilibrium community.
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0.2
0
0.4
0.2
0
10
20
30
1
0.6
(e)
0.4
0.2
0
10
20
0.6
10
20
30
0.2
0
1
0.4
0.2
10
20
10
20
30
Realized species richness
0.6
15
10
5
0
30
(g)
0.4
0.2
0
1
10
20
30
Realized species richness
(d)
20
1
Initial species richness
(f)
1
(c)
0.4
30
0
1
Realized species richness
0.6
Initial species richness
Minimum invariability
Initial species richness
Population invariability
1
Asymptotic resilience
(b)
Ecosystem invariability
0.4
0.6
10
20
30
Initial species richness
Ecosystem invariability
(a)
Population invariability
0.6
Minimum invariability
Asymptotic resilience
Figure S4
(h)
20
15
10
5
0
1
10
20
30
Realized species richness
Figure S4: Effect of absent species in random interactions model. Stability is plotted against
species richness for the model with random competitive interactions of Fig. 4. Results are
shown for community matrices A with absent species (×-marks) and without absent species
(circles). For each species richness level stability values are averaged over 1000 simulations.
(a−d) Initial species richness, i.e., the number of species in the species pool, is plotted on
the x-axis. (e−h) Realized species richness, i.e., the number of species in the equilibrium
community, is plotted on the x-axis.
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